3. Exploring numbers that yield squares and rectangles

When I give this lesson during an artist-in-residency presentation, the children are seated in twos or fours and supplied with 2-4 colors to be shared. I am at the front of the room with my overhead projector and some felt tip colored markers to use on a transparency showing the square grid.

I ask the children to color in one of the smallest squares on the grid. I tell them they can put that square anywhere they like on the paper. I say,

"Now that's a 1-square square.I walk around and look at the squares that are colored in. I will see squares that have 4 squares colored in, 9 squares, 16, etc. I will also see rectangles."Let's take another color marker and color in a bigger square. We don't want it to touch our 1-square square. How many squares must we color in to make a bigger square? Don't tell me; show me."

"Boys and girls, let's make the squares in size order. After the 1-square square, what is the next size square?"To facilitate the exploration of 'square numbers' I will use poker chips on the overhead projector and arrange 4 of them 'to make a square'. I will ask a student to come up to the overhead and add poker chips to make the 4-square into a bigger square. Before the student does each successive size, I will ask, "how many must we add?"

On the blackboard, two charts are being built as we go along:

Sequence # of Squares Add Yields alternating of Squares in the Square Squares odd and even #s 1st square 1-sq. square 3 even 2nd square 4-sq. square 5 odd 3rd square 9-sq. square 7 even 4th square 16-sq. square 9 odd 5th square 25-sq. square 11 even 6th square 36-sq. square 13 odd 7th square 49-sq. square 15 even 8th square 64-sq. square etc. 9th square 81-sq. square 10th square 100-sq. squareThus there are patterns of odd and even to help us know if the number of unit squares we think must be added to form a larger composite square is right, and the nth row of the 'Add Squares' column is equal to the number of squares you have to add to form the nth square from the n-1st one.

As we color in the successive unit squares we see a pattern on the grid. Each bigger composite square increases in width and height by 1. We count one more horizontal square, and a corresponding increase in vertical squares. The 1st square has 1 square, the 2nd has 2 squares, the 3rd 3, and, so on.

Unlike squares, rectangles have different numbers of squares in the vertical direction than the horizontal direction. Numbers that yield squares also yield rectangles: 9, for instance, makes a 3x3 square *and* a 9x1 rectangle.

Students who are members of the **'Secret Square Society'** know and can give me upon my asking a 'square number'. They will be able to visualize a rectangle by the number! They will know that certain numbers can be made into more than one rectangle. Children enjoy being in on the secret - and the game doesn't end until everyone gets into the **Secret Square Society**.

I point out to teachers that the numbers that do not make a rectangle or a square on the grid are **prime** numbers.

Coloring square grids can be very useful in:

- developing one's powers to visualize number patterns,
- understanding factoring and multiplication,
- developing concepts of perimeter and area,
- identifying prime numbers,
- doing everyday arithmetic operations.

Square grids make math artful. Students can make designs with squares that become

- letters of the alphabet,
- pictures,
- regular repeat patterns,
- original designs for needlepoint,

Students who are more visual than verbal may find these experiences especially meaningful.

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